Definition 2.2.1

Recall that an element in a C-star-Algebra is a projection if The set of projections in a -algebra is denoted by . From paragraph 2.1.1 we have the homotopy equivalence relation on . We introduce two additional equivalence relations on :

  1. if there exists such that and
    We call p a source projection, and we call q a range projection
    (We call this type of equivalence, Murray-von Neumann equivalence, or MvN equiv.)
  2. if there exists a unitary where is the Unitizations of , and we have that .
    Remark:
    When is a projection, we call it a partial isometry.
    If is a projection, then so is .
    We call a support projection. We call a range projection.
    Put and . Then we have the following much used identities:(see exercise 2.5)